The main purpose of the present paper is the heuristic study of the structure, properties and consequences of new class of potential functions results from the concept of pq-Radial functions which are fundamental family of solutions of second order pq-PDE. Keyword: Potential functions, Laplace equation, pq-Radial function, pq-PDE, Legendre polynomials

The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived using pole locations. Then, based on the Small-Gain theorem and on the Quadratic Separation framework, some robust stability criteria are provided. The latter f...

A Legendre spectral Galerkin method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of two Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplaci...

Weinstein’s[2] brilliant short proof of de Branges’[1] theorem can be made yet much shorter(modulo routine calculations), completely elementary (modulo Löwner theory), self contained(no need for the esoteric Legendre polynomials’ addition theorem), and motivated(ditto), as follows. Replace the text between p. 62, line 7 and p. 63, line 7, by Fact 1 below, and the text between the last line of p...

Time series problems involve analysis of periodic functions for predicting the future. A flexible regression method should be able to dynamically select the appropriate model to fit the available data. In this paper, we present a function approximation scheme that can be used for modeling periodic functions using a series of orthogonal polynomials, named Chebychev polynomials. In our approach, ...

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,

Newman polynomials are those with all coefficients in {0, 1}. We consider here the problem of finding Newman polynomials P such that all the coefficients of P 2 are so small as possible for deg P and P (1) given. A set A ⊂ [1, N ] is called a B2[g] sequence if every integer n has at most g distinct representations as n = a1 + a2 with a1, a2 ∈ A and a1 ≤ a2. Gang Yu [4] introduced a new idea to ...

Special cases of this theorem have been proved by Erdös-Grünwald' and Webster2 (the cases a = 1/2 and a = 3/2) . If there is no danger of confusion we shall omit the upper index n in lk`, n ~ (x) . PROOF OF THE THEOREM . It clearly suffices to consider the lk(x) with -1 =< xk < 0 . From the differential equation of the ultraspherical polynomials' we obtain (") lk(xk) = {«~xk) _ axk z zP„ (xk) 1...

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...

We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form ...